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A049417
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a(n) = isigma(n): sum of infinitary divisors of n.
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84
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1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 51, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 85, 84, 144, 68, 90
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OFFSET
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1,2
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COMMENTS
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A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.
Multiplicative: If e = Sum_{k >= 0} d_k 2^k (binary representation of e), then a(p^e) = Product_{k >= 0} (p^(2^k*{d_k+1}) - 1)/(p^(2^k) - 1). - Christian G. Bower and Mitch Harris, May 20 2005 [This means there is a factor p^2^k + 1 if d_k = 1, else the factor is 1. - M. F. Hasler, Oct 20 2022]
This sequence is an infinitary analog of the Dedekind psi function A001615. Indeed, a(n) = Product_{q in Q_n}(q+1) = n*Product_{q in Q_n} (1+1/q), where {q} are terms of A050376 and Q_n is the set of distinct q's whose product is n. - Vladimir Shevelev, Apr 01 2014
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LINKS
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FORMULA
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Let n = Product(q_i) where {q_i} is a set of distinct terms of A050376. Then a(n) = Product(q_i + 1). - Vladimir Shevelev, Feb 19 2011
Multiplicative with a(p^e) = Product{k >= 0, e_k = 1} p^2^k + 1, where e = Sum e_k 2^k, i.e., e_k is bit k of e. - M. F. Hasler, Oct 20 2022
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EXAMPLE
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If n = 8: 8 = 2^3 = 2^"11" (writing 3 in binary) so the infinitary divisors are 2^"00" = 1, 2^"01" = 2, 2^"10" = 4 and 2^"11" = 8; so a(8) = 1+2+4+8 = 15.
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MAPLE
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isidiv := proc(d, n)
local n2, d2, p, j;
if n mod d <> 0 then
return false;
end if;
for p in numtheory[factorset](n) do
padic[ordp](n, p) ;
n2 := convert(%, base, 2) ;
padic[ordp](d, p) ;
d2 := convert(%, base, 2) ;
for j from 1 to nops(d2) do
if op(j, n2) = 0 and op(j, d2) <> 0 then
return false;
end if;
end do:
end do;
return true;
end proc:
idivisors := proc(n)
local a, d;
a := {} ;
for d in numtheory[divisors](n) do
if isidiv(d, n) then
a := a union {d} ;
end if;
end do:
a ;
end proc:
local d;
add(d, d=idivisors(n)) ;
end proc:
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MATHEMATICA
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bitty[k_] := Union[Flatten[Outer[Plus, Sequence @@ ({0, #1} & ) /@ Union[2^Range[0, Floor[Log[2, k]]]*Reverse[IntegerDigits[k, 2]]]]]]; Table[Plus@@((Times @@ (First[it]^(#1 /. z -> List)) & ) /@ Flatten[Outer[z, Sequence @@ bitty /@ Last[it = Transpose[FactorInteger[k]]], 1]]), {k, 2, 120}]
(* Second program: *)
a[n_] := If[n == 1, 1, Sort @ Flatten @ Outer[ Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] // Total;
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PROG
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(PARI) A049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)))} \\ Andrew Lelechenko, Apr 22 2014
(PARI) isigma(n)=vecprod([vecprod([f[1]^2^k+1|k<-[0..exponent(f[2])], bittest(f[2], k)])|f<-factor(n)~]) \\ M. F. Hasler, Oct 20 2022
(Haskell)
a049417 1 = 1
a049417 n = product $ zipWith f (a027748_row n) (a124010_row n) where
f p e = product $ zipWith div
(map (subtract 1 . (p ^)) $
zipWith (*) a000079_list $ map (+ 1) $ a030308_row e)
(map (subtract 1 . (p ^)) a000079_list)
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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